Hidden Variable Fractal Interpolation Functions

Barnsley, Michael F.; Elton, John H.; Hardin, Douglas P.; Massopust, Peter

doi:10.1137/0520080

Cited by 181 publications

(86 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…To model a function or a data set with non-self-referential nature Barnsley et al [20] initiated the notion of hidden variable FIFs. For simulating curves that exhibit partly self-referential and partly non-self-referential nature, coalescence hidden variable FIF is introduced in [21].…”

Section: Fractal Interpolation Theory: An Overviewmentioning

confidence: 99%

Fractal Interpolation Functions: A Short Survey

Navascués¹,

Chand²,

Viswanathan³

et al. 2014

View full text Add to dashboard Cite

The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.

show abstract

Section: Fractal Interpolation Theory: An Overviewmentioning

confidence: 99%

Fractal Interpolation Functions: A Short Survey

Navascués¹,

Chand²,

Viswanathan³

et al. 2014

View full text Add to dashboard Cite

show abstract

“…In view of their diverse applications, there has been steadily increasing interest in the theory of fractal functions, and it still continues to be a hot topic of research. Following the publication of Fractals Everywhere [2], a beautiful exposition of IFS theory, fractal functions and their applications, various related issues such as calculus, Holder continuity, convergence, stability, smoothness, determination of scaling parameters, and perturbation error have been investigated in the literature [3][4][5][6][7][8][9][10][11][12][13]. The concept of smooth FIFs has been used to generalize the traditional splines [14][15][16][17][18] and to demonstrate that the interaction of classical numerical methods with fractal theory provides new interpolation schemes that supplement the existing ones.…”

Section: Prologuementioning

confidence: 99%

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

AKB¹

2014

J Appl Computat Math

View full text Add to dashboard Cite

This article is targeted to provide a brief and coarse discussion on theory of fractal functions and their recent developments including some of the researches of the authors. Materials on fractal functions provided by this paper is not claimed to be exhaustive, but is intended to be read before or in parallel with technical papers available in the literature on this subject. We have made an earnest attempt to supply an overall flavor of the subject of univarite fractal approximation and useful source of links to assist a novice and perhaps to an expert in fractal functions too.

show abstract

“…Unfortunately this is not the case for the maps considered here since the condition is equivalent to T 2 α < 0. However, by using an equivalent metric (used in [Glendinning, 2014] for the contracting case and [Barnsley et al, 1989] for Iterated Function Systems) we are able to find a metric in which the affine maps can be expanding, and hence the results of Tsujii [2001] and Buzzi [2001] can be used to establish the existence of an absolutely continuous invariant measure and, generically, an attractor with non-empty interior. Explicit regions of parameter space are identified in the proof.…”

Section: Introductionmentioning

confidence: 99%

Invariant Measures for the n-Dimensional Border Collision Normal Form

Glendinning

2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

The border collision normal form is a continuous piecewise affine map of R n with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence that the attractors have Hausdorff (fractal) dimension n. If n = 2 the attractors have topological dimension two, i.e. they contain open sets, and if n > 2 then they have topological dimension n generically.

show abstract

Hidden Variable Fractal Interpolation Functions

Cited by 181 publications

References 7 publications

Fractal Interpolation Functions: A Short Survey

Fractal Interpolation Functions: A Short Survey

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

Invariant Measures for the n-Dimensional Border Collision Normal Form

Contact Info

Product

Resources

About